SCS-CN hydrographs

The SCS-CN method (Soil Conservation Service, now NRCS) transforms a design hyetograph into the flood hydrograph at a basin section. Two steps:

1. Net rainfall from total rainfall via the Curve Number;
2. Convolution with the SCS unit hydrograph, scaled to the lag time.

Step 1 — net rainfall

The model assumes part of the rainfall is absorbed (initial abstraction \(I_a\)) or accumulates as potential retention \(S\):

\[ S = \frac{25400}{\text{CN}} - 254 \qquad I_a = 0.2 \cdot S \]

With \(S\) in mm and CN in \([0, 100]\). For total cumulative rainfall \(P\):

\[ P_e = \begin{cases} 0 & \text{if } P \le I_a \\[1mm] \dfrac{(P - I_a)^2}{P - I_a + S} & \text{if } P > I_a \end{cases} \]

\(P_e\) is the cumulative net rainfall. Initial losses represent vegetal interception + surface storage. Below \(I_a\) everything infiltrates.

Curve Number

CN depends on the hydrologic soil type (A, B, C, D — from highly permeable to impermeable) and land cover. Examples:

Cover	A	B	C	D
Moderately covered woods	30	55	70	77
Permanent meadows	39	61	74	80
Clean row crops	67	78	85	89
Impervious urban areas	98	98	98	98

Integrated CN wizard

In the hydrograph panel an interactive wizard lets you: 1. split the basin into multiple sub-areas; 2. assign hydrologic soil and land cover to each; 3. get the area-weighted CN for the basin.

See the SCS Tables section for the complete tables.

Step 2 — SCS unit hydrograph

The dimensionless SCS unit hydrograph is a triangle with:

• time to peak \(t_p = 0.5\,D + T_{\text{lag}}\), where \(D\) is the net rainfall duration and \(T_{\text{lag}}\) the basin lag time;
• unit peak discharge \(q_p = 484 \cdot A / t_p\) (US units; in SI: \(q_p = 0.208 \cdot A / t_p\) with A in km² and \(t_p\) in hours);
• base time \(t_b = 2.67 \cdot t_p\).

The total hydrograph is the convolution of the \(\Delta P_e(t)\) sequence with the unit hydrograph:

\[ Q(t) = \sum_{k} \Delta P_e(k) \cdot u(t - k) \]

Lag time — \(T_{\text{lag}}\)

\(T_{\text{lag}}\) is the time between the centroid of net rainfall and the hydrograph peak. Typical estimates:

• SCS formula: \(T_{\text{lag}} = 0.6 \cdot T_c\) (Tc = time of concentration).
• Time of concentration: Kirpich, Giandotti, Pasini, Pezzoli depending on basin morphometry.

Tlag wizard

In the panel, the Tlag calculator accepts: - basin area A (km²); - main channel length L (km); - average slope i (%);

It returns \(T_c\) with four classical formulas + the suggested \(T_{\text{lag}}\) as 0.6 \(T_c\). Choose the value you judge best suited to your basin type.

Using it in Runoff Lab

1. Go to Hydrographs panel → Add.
2. Source: a synthetic hyetograph already built (or a direct curve).
3. Enter:
4. basin area A (km²);
5. CN (from wizard or manual);
6. Tlag (from wizard or manual).
7. Confirm → \(Q(t)\) chart, table with:
8. \(Q_p\) (peak discharge, m³/s);
9. \(t_p\) (time to peak, h);
10. \(V_{\text{tot}}\) (total runoff volume, m³);
11. \(P_e\) (total net rainfall, mm);
12. net rainfall / gross rainfall ratio (runoff ratio).

SCS-CN method limitations

• Originally developed for small US agricultural basins (≤ 250 km²). For large basins, a distributed model is preferable.
• The assumption \(I_a = 0.2 S\) is a simplification: for highly impervious urban basins, \(I_a = 0.05 S\) may be more appropriate.
• Constant (linear) Tlag — doesn't capture loss non-linearities for extreme floods.

Bibliography

• Soil Conservation Service (1972, 1986), National Engineering Handbook, Section 4: Hydrology. USDA.
• Mockus V. (1957), Estimation of total (and peak rates of) surface runoff for individual storms, USDA.
• Maidment D.R. (ed., 1993), Handbook of Hydrology, McGraw-Hill — ch. 9 (SCS).

Full CN tables

(For brevity, an excerpt here; the in-app wizard has complete tables for: woods, meadows, row crops, urban areas, residential, commercial, infrastructure, natural bare ground.)